Open Access

Sharpening the Becker-Stark Inequalities

Journal of Inequalities and Applications20102010:931275

https://doi.org/10.1155/2010/931275

Received: 3 April 2009

Accepted: 14 January 2010

Published: 20 January 2010

Abstract

In this paper, we establish a general refinement of the Becker-Stark inequalities by using the power series expansion of the tangent function via Bernoulli numbers and the property of a function involving Riemann's zeta one.

1. Introduction

Steckin [1] (or see Mitrinovic [2, 3.4.19, page 246]) gives us a result as follows.

Theorem 1.1 (see [1, Lemma ]).

If , then
(11)

Later, Becker and Stark [3] (or see Kuang [4, 5.1.102, page 248]) obtain the following two-sided rational approximation for .

Theorem 1.2.

Let , then
(12)

Furthermore, and are the best constants in (1.2).

In fact, we can obtain the following further results.

Theorem 1.3.

Let , then
(13)

Furthermore, and are the best constants in (1.3).

In this paper, in the form of (1.2) and (1.3) we shall show a general refinement of the Becker-Stark inequalities as follows.

Theorem 1.4.

Let , and let be a natural number. Then
(14)
holds, where , and
(15)

where are the even-indexed Bernoulli numbers.

Furthermore, and are the best constants in (1.4).

2. Four Lemmas

Lemma 2.1.

The function is decreasing, where is Riemann's zeta function.

Proof.

is equivalent to the function , which is decreasing.

Lemma 2.2 (see [5, Theorem ]).

Let be Riemann's zeta function and the even-indexed Bernoulli numbers. Then
(21)

Lemma 2.3 (see [6, 1.3.1.4 (1.3)]).

Let . Then
(22)

Lemma 2.4.

Let and . Then , where
(23)

Proof.

By Lemma 2.3, we have
(24)
Since is decreasing by Lemma 2.1, it follows that
(25)
From Lemma 2.2, we get
(26)

which implies that for .

3. Proofs of Theorems

Proof of Theorem 1.4.

Let
(31)
Then
(32)

By Lemma 2.4, we have for , and is decreasing on .

At the same time, = by (3.1), and by (3.2), so and are the best constants in (1.4).

Proof of Theorem 1.3.

Let in Theorem 1.4; we obtain that and . Then the proof of Theorem 1.3 is complete.

Authors’ Affiliations

(1)
Department of Mathematics, Zhejiang Gongshang University

References

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Copyright

© L. Zhu and J. Hua 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.